This question is asked in the context of the famous paper on DFT by Hohenberg and Kohn. What makes the inhomogeneous electron gas more interesting than its homogeneous counterpart?
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2 Answers
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The homogeneous electron gas is a fiction: it represents a system that is completely uniform and that has a constant, smooth positive background charge to make the system charge neutral.
However, the homogeneous electron gas is useful: since the electron density is the same at every point in space, there is a simple mapping from the density to the exact energy of the system. For instance, you can evaluate the exact energy with e.g. Quantum Monte Carlo for every value of the density, and then interpolate between the values.
Real molecules and crystals do not have a homogeneous external field. Instead, the Coulomb potential of atomic nuclei is highly inhomogeneous. This is why the inhomogeneous electron gas is important in practice, and it is also more challenging to model than the theoretical homogeneous electron gas.
The theories for the inhomogeneous electron gas, such as generalized gradient approximations (GGAs) and meta-GGAs, are commonly built as corrections to the LDA. For instance, exchange functionals are often written as $$ E_{\rm xc} = \int n \epsilon^{\rm LDA}_{\rm xc} (n) F_x (\nabla n, \tau) {\rm d}^3r $$ where $F_x$ is the enhancement function which modifies the LDA exchange energy density $n \epsilon^{\rm LDA}$ depending on the gradient and local kinetic energy in the point, for a meta-GGA.
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1$\begingroup$ Thank you for the answer @Dr Susi Lehtola. Yet, I am failing to see the use of homgeneous electron gas concept apart from LDA approximation wherein the exchange-correlation potential of actual inhomogeneous electron gas is approximated by that of homogenous counterpart at that local point. In subsequent approximations e.g. GGA, the derevatives of varying ( and not constant or uniform) density is considered. Please elaborate on this. $\endgroup$– AbPhysCommented Mar 7, 2023 at 19:02
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$\begingroup$ @AbPhys I amended the answer. $\endgroup$ Commented Mar 8, 2023 at 12:04
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$\begingroup$ Edited answer now helped me get the point. Reading this, one more question popped up in mind- Is this why the meta-GGA approximations e.g. Tran-Blaha modified BJ XC potential fall under the category of model Hamiltonians? $\endgroup$– AbPhysCommented Mar 9, 2023 at 15:39
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$\begingroup$ @AbPhys all density functional approximations are models. $\endgroup$ Commented Mar 9, 2023 at 17:14
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In contemporary Kohn-Sham DFT, the exchange and correlation energies are local quantities. If the electron density is slowly varying, the LDA form derived from a uniform electron gas is a zero'th order approximation. This is the underlying reason that the various XC functionals written in terms of density, density gradient, and possibly higher order information, often are required to revert to the LDA form when the derivatives become zero.
